Communications in Mathematical Sciences

Volume 11 (2013)

Number 1

Slow manifolds for multi-time-scale stochastic evolutionary systems

Pages: 141 – 162



Jinqiao Duan (Institute for Pure and Applied Mathematics, University of California at Los Angeles)

Hongbo Fu (College of Mathematics and Computer Science, Wuhan Textile University, Wuhan, China)

Xianming Liu (School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, China)


This article deals with invariant manifolds for infinite dimensional random dynamical systems with different time scales. Such a random system is generated by a coupled system of fast- slow stochastic evolutionary equations. Under suitable conditions, it is proved that an exponentially tracking random invariant manifold exists, eliminating the fast motion for this coupled system. It is further shown that if the scaling parameter tends to zero, the invariant manifold tends to a slow manifold which captures long time dynamics. For illustration, the results are applied to a few systems of coupled parabolic-hyperbolic partial differential equations, coupled parabolic partial differential-ordinary differential equations, and coupled hyperbolic-hyperbolic partial differential equations.


stochastic partial differential equations (SPDEs), random dynamical systems, multiscale systems, random invariant manifolds, slow manifolds, exponential tracking property

2010 Mathematics Subject Classification

37D10, 37L55, 60H15, 70K70

Published 7 September 2012